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Bertini's Theorem on Generic Smoothness

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Bertini's Theorem on Generic Smoothness U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´ Master Thesis in Mathematics BERTINI1S THEOREM ON GENERIC SMOOTHNESS Academic year 2011/2012 Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present disser- tation concerns his most celebrated theorem, which appeared for the first time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its re- cent improvements and its relationships with other kind of re- sults. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following: ¥ there were no schemes, ¥ almost all varieties were defined over the complex numbers, ¥ all varieties were embedded in some projective space, that is, they were not intrinsic. On the contrary, this dissertation will cope with Bertini’s the- orem by exploiting the powerful tools of modern algebraic ge- ometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our vari- eties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave. By the way, iii iv the interested reader can find a detailed description of the origi- nal Bertini’s theorems in the beautiful and accurate paper [15] by S. L. Kleiman. Unless otherwise specified we will work with varieties over an arbitrary algebraically closed field k. Bertini’s theorem as- serts (in its weak form) that the generic hyperplane section of a smooth projective variety preserves the original smoothness; moreover, the set of hyperplanes satisfying this condition forms itself a quasiprojective variety (a behavior recalling us that param- eter spaces of algebraic varieties are usually algebraic varieties themselves). For convenience, we informally say that a (local) property Q of k-varieties is Bertini (or weak-Bertini) if it satisfies condition (B) below and it is strong-Bertini if it satisfies condition (SB): n (B): If X Pk is a variety satisfying Q then the general hyperplane section of X satisfies Q. (SB): Let X be a variety with a finite dimensional linear system S on it and let X 99K S be the corresponding rational map. If this map induces separable extensions on residue fields (of closed points), then the generic member of S satisfies Q, away from the base points of S and the points in X that are not Q. Our aim is to show that Q smoothness is (B) and to con- vince ourselves, with some examples rather than by way of pre- cise proofs, that it is also (SB). For most of the time, however, we will be concerned with the weaker (B), so we will become familiar with hyperplane sections in any characteristic; those hy- perplanes H giving a smooth intersection with X will be called good, otherwise they will be called bad. We will also show that in characteristic zero smoothness is (SB) (note that the condition of separability is automatically satisfied). With an explicit exam- ple and a reference to a theorem of Kleiman, we hope to clarify that the inseparability of the extensions between residue fields is exactly the obstruction for smoothness to become (SB) in positive characteristic. Let us precise that smoothness is not the unique ”Bertini type” property. Suppose, indeed, you have a property Q on your variety X Pn. Then you may ask yourself several questions: 1. Can I find at least one hyperplane H such that XXH satisfies Q? (sometimes no such one exists) v 2. How many H are there such that X X H satisfies Q? (some- times few such exist) 3. Does the answer to the previous questions depend on the characteristic of the base field? (sometimes positive charac- teristic makes life hard) Here are examples of Q for which one knows that (B) holds: being reduced, irreducible, normal, Cohen-Macaulay, smooth; and of course it is very interesting to try to enlarge this list as much as possible. It is noteworthy that a property Q has a remarkable hope of being (B) if it is constructible (and the base field is not too small - finite, for example). This will be made clear later but, for the moment, just observe that a constructible set of an irreducible Zariski space either contains an open subset, or is nowhere dense. In what follows we attempt to give a quick description of some more recent results regarding Bertini’s Theorem in its sev- eral materializations. Some of them will be discussed later in this work, while others just appear in this introduction as a cue for the reader. The main directions along with the theorem can be improved are: I. Bertini theorems over Fq; II. study on new Bertini properties; III. deduce some new results in Algebraic Geometry; IV. Bertini over a ring of integers. Let us analyze each of them a little more carefully. § I. The problem with finite fields. With finite fields, everything is finite! The number of rational points in the projective space, n the set of hyperplanes... When k is any field, and X Pk is a smooth quasiprojective variety of dimension m ¥ 0, then one may consider the set t P n¦ | X ¡ p q u U u Pk X Hu is smooth of dimension m 1 over k u n where Hu Pkpuq is the hyperplane corresponding to the point u, defined over the residue field kpuq. It turns out that U is dense n¦ in Pk . If k is infinite, then one can find a hyperplane in U which is defined over k. Otherwise, if k Fq, all the finitely many hyperplanes H over k might give a singular intersection X X H. It is a question by N. M. Katz to see what happens if one considers hypersurfaces of degree greater than one, instead of just looking vi r s at hyperplanes. Let S be the homogeneous ring Fq x0,..., xn of n P P , and for every homogeneous f Sh d¥0 Sd let Hf be the hypersurface Proj S{f Pn. Define the density 7 P X Sd µpPq lim dÑ8 7 Sd for any subset P Sh. Then Poonen in [18] (2004), answering to Katz’s question, proves that if X Pn is smooth and quasiprojec- tive of dimension m ¥ 0 over Fq, then the subset P t f P Sh | X X Hf is smooth of dimension m ¡ 1 u ¡1 has positive density, equal to ζXpm 1q , where ζX is the zeta function attached to the variety X. So, Poonen makes Bertini’s statement become true over Fq by allowing hypersurfaces to play the role of classical hyperplanes. Another interesting result in characteristic p ¡ 0 is due to E. Ballico. In 2003 he proves (see [4]) what follows. He lets k be the algebraic closure of Fp, but in fact he focuses on good n hyperplanes H Pk defined over Fq, where q is a power of p. n In his theorem, X Pk is an irreducible variety of dimension m and degree d. He proves that if q ¥ dpd ¡ 1qm then there exists n 1 some hyperplane H Pk defined over Fq and transversal to X. Note that X might not be defined over Fq. However, the result still holds if X is smooth and defined over Fq: there still exists a good hyperplane cutting X transversally at all of its k-points (and not just at Fq-points): this means that for every x P X one has TX,x H. § II. New Bertini properties and generalizations. Let us first fo- cus on the problem of finding new Bertini properties. There exist two properties that we will deal with in this paragraph: weak- normality (WN), and property WN1. Here and in the follow- ing a reduced variety X will be said to be WN if every birational universal homeomorphism Y Ñ X is an isomorphism.2 Instead, WN1 means WN with the extra condition that the normalization morphism X˜ Ñ X is unramified in codimension one. One al- ways has that WN1 ñ WN, and they agree in characteristic zero 1 R ¦ n¦ This means that H X Pk . If X is smooth, H is transversal to X if and P only if TX,x H for all x X. If it is not smooth and H cuts it transversally, then P TX,x H for all x Xsm. 2 Equivalently, X coincides with its weak normalization, the maximal couple pXˆ, fˆq among birational universal homeomorphisms Y Ñ X. A universal homeo- morphism is a Zariski homeomorphism f such that for every y P Y the extension kpfpyqq kpyq is purely inseparable (so it is trivial when char k 0). vii or for varieties of dimension one. Cumino, Greco and Manaresi showed (see [6]) that Q WN is (B) for varieties of every dimen- sion in characteristic zero. By means of an ”axiomatic approach” to Bertini properties, the same authors show (see [7]) that in fact WN is (SB), again in characteristic zero. The strategy is as follows.
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